Fridays 4:10pm, Stevenson Center 1307

Abstract: A general historical perspective is traced for Harnack-type inequalities in degenerate and/or singular quasi-linear evolution equations. The outstanding open issues of the 60’s arising from the work of Moser, Serrin and Trudinger, will be recalled and discussed. The recent solutions of these long standing problem will be presented in terms of the new idea of “intrinsic geometry”.Quasi-linear parabolic equations do not admit a “potential” nor comparison principles in any sense. We introduce the notion of a “generalized sub-potential” and its existence will be traced to the possibility of a Harnack estimate. Thus Harnack is possible if and only if a degenerate quasi-linear PDE admits a “sub-potential”.

Abstract: We consider quasilinear systems of quasilinear parabolic partial differential equations with fully nonlinear boundary conditions, say on a bounded domain, under standard ellipticity conditions. We first establish local wellposedness and smoothing properties of the solutions. In the main theorems we investigate the qualitative behavior near an equilibrium: We construct the local stable, center and unstable manifolds, derive their fundamental properties such as local invariance, and study their impact on the asymptotic behavior.Our results are based on fixed point methods, maximal $L_p$-regularity for linear initial-boundary value problems, semigroup theory, and the behavior of substitution operators on certain anisotropic Slobodetskii spaces. The solutions live on a nonlinear manifold defined by the boundary conditions. This fact leads to several crucial new difficulties which play a major role in our arguments.

Date: Friday, 5 October 2007

Speaker: Vincenzo Vespri, University of Florence, Italy

Title: Continuity of the Saturation in the Flow of Two Immiscible Fluids Through a Porous Medium

Abstract: We consider a weakly coupled, highly degenerate system of an elliptic equation and a parabolic equation, arising in the theory of flow of immiscible fluids in a porous medium. The unknown functions in the system are $u$ and $v$, and represent pressure and saturation, respectively. By using suitable DeGiorgi’s techniques, we prove that the saturation is a continuous function in the space-time domain of definition. Here the main technical point is that no assumptions at all are made on the nature of the degeneracy of the system.

Abstract: In this talk the following problems will be introduced and discussed:
1. Stationary forms of rotating liquids.
2. Hydrodynamic free boundary problems.

Date: Thursday, 25 October 2007, Colloquium talk (4pm, SC 5211)

Speaker: Konstantina Trivisa, University of Maryland

Title: Multicomponent reactive flows

Abstract: Multicomponent reactive flows arise in many practical applications such as combustion, atmospheric modelling, astrophysics, chemical reactions, mathematical biology etc. The objective of this work is to develop a rigorous mathematical theory based on the principles of continuum mechanics.

Date: Friday, 26 October 2007

Speaker: Konstantina Trivisa, University of Maryland

Title: From the dynamics of gaseous stars to the incompressible Euler equations

Abstract: A model for the dynamics of gaseous stars is introduced formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier-Stokes-Poisson system in the torus is investigated. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.

Abstract: The lecture will discuss paradigms from the theory of hyperbolic conservation laws in which physics imposes special challenges to the analyst, but then also provides the means of overcoming these challenges.

Abstract: We consider the Cauchy problem for a doubly nonlinear second order parabolic equation, involving a first-order term (i.e., the derivative of a power of the unknown with respect to a space variable). We investigate the asymptotic properties of nonnegative solutions for large times. Specifically, we find optimal estimates for the sup norm of the solution, and for its (compact) support. Such estimates depend sharply on the relative size of the parameters appearing in the equation.

Abstract: Infections caused by antibiotic-resistant pathogens in hospitals are an increasing global health problem. This rise reflects an incomplete understanding of the complexities of transmission dynamics and antibiotic exposure in patient-healthcare worker contamination and healthcare worker-patient colonization. Mathematical models are developed to assess the emergence and spread of antibiotic resistance in hospitals. Analysis of these models demonstrates that early initiation of treatment and minimization of its duration mitigates antibiotic resistance epidemics in hospitals.

Date: Friday, 7 December 2007

Speaker: Borys Bazaliy, Academy of Science, Donetsk, Ukraine

Title: Free boundary problems in mathematical physics II

Abstract: This talk will be a continuation of the talk given on October 19. The following problems will be introduced and discussed:
1. Problems in phase transitions.
2. Nonlinear filtration problems.
3. Some biophysical free boundary problems.

Abstract: For an elliptic partial differential equation with coefficients that vary on a very fine scale relative to the domain size, standard finite element methods are not practical due to the largeness of the resulting problem. The type of equation arises, for example, in the study of fluid flow in a heterogeneous porous medium. Tom Hou et al. defined particular “multiscale” basis functions for use in the numerical solution of such equations; independently, Todd Arbogast et al. developed an abstract “subgrid upscaling” framework for the study of the interaction of scales in these problems. In a recent paper, Arbogast and the speaker have placed Hou’s work in this framework, leading to a different proof of the error estimates, as well as to extensions of the results.

Date: Friday, February 15, 2008, 4:10pm, SC1307

Speaker: Mikhail Perepelitsa, Vanderbilt University

Title: Some problems in the theory of Compressible Navier-Stokes Equations

Abstract: We will overview the basic facts in the existence/regularity theory of the Navier-Stokes equations for compressible flows. The theory appears to be almost complete in 1 dimension, while in higher dimensions there are fundamental open issues. We will discuss in details weak solutions that have a piece-wise smooth structure and the stability of the equilibrium solutions within this class.

Abstract: Often the most important solutions of parabolic partial differential equations are positive. But if the equation contains a term that is too singular, positive solutions may fail to exist. We explain this, including the mechanism that causes “instantaneous blow up”. The classical linear results are in Euclidean space and on the Heisenberg group, with a singular potential with special scaling properties. In nonlinear versions the Laplacian may be replaced by the p-Laplacian or the porous medium operator, and there is now the beginning of a theory on Carnot groups. The lecture should be accessible to those interested in analysis who do not know many of the above terms.

Date: Friday, February 29, 2008

Speaker: John Graef, University of Tennessee at Chattanooga

Title: Nonlinear Boundary Value Problem for a Wing in an Air Flow or Why Engineers Should Use Nonlinear Analysis

Abstract: We consider the problem of modeling the torsion of a wing in an air flow. An approximate equation for the torsion is derived. Along with the boundary conditions, this forms a non-linear Sturm-Liouville problem with the Mach number as the spectral parameter. Conditions under which a unique solution of the problem exists are presented and a characterization of the corresponding (smallest) Mach number is given. This smallest eigenvalue, which leads to the failure of the wing, is estimated by using a cubic approximation. The techniques used here also allow the calculation of the exact value for dry air. For some values of the parameters, these are both found to be significantly smaller than the value obtained using standard linear approximation techniques. (This is joint work with B. P. Belinskiy and R. E. Melnik of the University of Tennessee at Chattanooga.)

Title: Qualitative behavior of solutions for the generalized Stefan problem with surface tension

Abstract: Strong local well-posedness for the generalized two-phase Stefan problem with surface tension and with or without kinetic undercooling is discussed. The only equilibria of this system are constant temperatures and finitely many non-intersecting spheres with equal radii. The stability of such equilbria is discussed by means of the generalized principle of linearized stability. We show that a solution which does not exhibit singularities is global and converges to one of these equilibria as time tends to infinity.

Date: Friday, March 21, 2008

Speaker: David Hoff, Indiana University

Title: Analyticity in time and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow

Abstract: We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. One important corollary is backwards uniqueness: if two such solutions agree at a given time, then they must agree at all previous times. Additionally, analyticity yields sharp estimates for the time derivatives of arbitrary order of solutions along particle trajectories.
I’m going to integrate into the talk something like a “pretalk,” in an attempt to motivate the more technical material and to make things accessible to general analysis grad students. All are welcome.

Abstract: We consider a linear stochastic wave eq-n driven by fractional-in-time noise. We prove the existence and uniqueness of the weak solution.We also study the expected energy associated with wave eq-n and improve our previous results on that matter. Specifically, we find the iff condition of the convergence of the series representing the expected energy using physically natural objects. We discuss the smoothness of the solution. We consider both cases H > 1/2 and H < 1/2 for the Hurst parameter.

Abstract: Blood clotting is the result of a large number of chemical reactions leading in particular to the formation of fibrin in the blood stream. Fibrin modifies the rheological properties of blood and at the same time it is used by platelets to build up a gel-like structure, enclosing plasma, red blood cells and other blood constituents. This is what we call a blood clot. On a small scale this process occur normally in the body in order to repair micro-wounds and is also needed in exceptional circumstances to repair large wounds. Unfortunately it may take place in a rather massive way leading to vessels obstruction either due to blood stagnation, or to a mechanical activation of platelets (i.e. the exposure of platelets to a stress beyond some threshold for a sufficiently long time) in combination with fibrin accumulation.We present a mathematical model referring to the latter situation. The key point of the model is to select a range of parameters guaranteeing a clear separation of the various time scales involved in the process (chemical reactions, diffusion of the chemicals, fluid dynamics, advancement of the clotting front from the vessels walls). Under such conditions the chemistry is described by a set of o.d.e.’s, the fluid dynamics is quasi-steady and the progression of clotting is driven by the motion of two free boundaries: a boundary of platelets activation and the boundary of clot formation.
The free boundary conditions are of an unusual type. Existence and uniqueness are proved and the consistency of the assumptions made on the time scales is investigated.